Disclaimer: This document was part of the First
European DSP Education and Research Conference.
It may have been written by someone whose native
language is not English. TI assumes no liability for the
quality of writing and/or the accuracy of the
information contained herein.
Sensorless Control of a Trapezoidal
Brushless DC Motor Using the
TMS320C25 DSP
Authors: S. Grainger, M. Allan, T. Westwood,
R. Champaneri
ESIEE, Paris
September 1996
SPRA323
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Contents
Abstract ........................................................................................................................... 7
Product Support on the World Wide Web .................................................................... 8
Background..................................................................................................................... 9
Software Algorithm......................................................................................................... 9
Simulation Results ....................................................................................................... 15
Experimental Setup ...................................................................................................... 16
Algorithm Implementation ........................................................................................... 18
Experimental Results ................................................................................................... 20
Summary ....................................................................................................................... 23
Future Work................................................................................................................... 23
References .................................................................................................................... 23
Figures
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Schematic Representation of the Algorithm ....................................................... 9
Algorithm Flowchart.......................................................................................... 13
Experimental Setup .......................................................................................... 17
Motor BEMF Magnitude.................................................................................... 20
Normalized BEMF ............................................................................................ 21
Permanent Magnet Flux Linkage...................................................................... 21
Calculated Position and Error........................................................................... 22
Sensorless Control of a Trapezoidal
Brushless DC Motor Using the
TMS320C25 DSP
Abstract
This application report presents the theory and practice of
implementing a closed-loop sensorless commutation scheme for a
trapezoidal brushless DC motor. The implementation consists of a
PC-hosted Texas Instruments (TITM) TMS320C25 digital signal
processor (DSP).
The TMS320C25 functions as a coprocessor running an algorithm
to determine the phase voltage switching points. This solution
uses the DSP as an alternative to the bulky, temperature
sensitive, hall-effect sensors traditionally used. The proposed
future works suggests the algorithms successfully used with the
TMS320C25 may yield even faster sampling rates with use of TI’s
TMS320C30 DSP.
This document was part of the first European DSP Education and
Research Conference that took place September 26 and 27, 1996
in Paris. For information on how TI encourages students from
around the world to find innovative ways to use DSPs, see TI’s
World Wide Web site at www.ti.com.
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
7
SPRA323
Product Support on the World Wide Web
Our World Wide Web site at www.ti.com contains the most up to
date product information, revisions, and additions. Users
registering with TI&ME can build custom information pages and
receive new product updates automatically via email.
8
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
Background
A brushless DC motor (BLDCM) is one in which the commutation
function normally performed by the brushes and commutator in a
traditional DC motor is replaced with an electronic equivalent. This
electronic circuit must switch phase voltages on and off at
particular instants related to the position of the motor’s rotor. A
trapezoidal BLDCM has a trapezoidal flux distribution and is
supplied with square wave voltages or currents.
For a 3-phase, 2-pole trapezoidal motor, as used in this case, a
switching point occurs every 60° but it is necessary to determine
this point to within approximately ±2°.1
Software Algorithm
For a motor running at speed, the rotor position is most easily
deduced from the motor’s back EMF (BEMF); that is, the voltage
induced in the phase windings by the action of the rotating
magnetic field. With a typical trapezoidal BLDCM there is always
one phase not energized. The voltage of this phase can be
measured directly. However, as this voltage is proportional to
speed, this method cannot be used for startup and slow speed
operation.
Ertugrul and Acarnley have proposed a more sophisticated
method of determining the rotor position as shown schematically
in Figure 1 and described below.1 2 Although they used this
algorithm to determine the rotor position of a sinusoidal BLDCM in
real time, they did not use it to control the motor commutation.
Figure 1. Schematic Representation of the Algorithm2
i1i
CURRENT
ESTIMATION
1
POSITION
CORRECTION
&
POSITION
ESTIMATION
&
PREDICTION
CURRENT
ESTIMATION
2
FLUX
LINKAGE
CORRECTION
2
i
3
v
1
v
2
v
3
∫
INTEGRATOR
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
9
SPRA323
A commonly used electrical motor model is:
vp = ipR +
dΨ( i
k
θ ,Ψ )
f
(1)
dt
where:
p
=
phase number
k
=
total of phases
vp
=
applied phase voltage
ip
=
phase current
R
=
phase resistance
Ψ
=
phase flux linkage
ik
=
current out of all phases
θ
=
rotor position
Ψf
=
flux linkage due to rotor time
t
=
time
Expressing this in terms of separate component parts:
{
v p = ip R +
I
∂Ψ dik ∂Ψ dθ
∂Ψ dΨ f
+
+
∂ik dt ∂θ dt ∂Ψ f dt
14243 123 14243
III
II
(2)
IV
where:
I
II
= resistive voltage drop
= inductive voltage drop
III
IV
= motional BEMF
= voltage change due to reluctance
Assuming negligible reluctance effect and negligible nonlinearities due to saturation, the coil and mutual inductances are
constant. Therefore:
*
*
*
v1 = i1 R + L1,1i1 + M 1,2 i1 + M 1,3i1 + e1
*
*
*
v 2 = i2 R + M 2,1i2 + L2,2 i2 + M 2,3i2 + e1
{ 14444244443
*
*
*
v 31 = i3 R + M 3,1i3 + M 3,2 i3 + L3,3i3 + e{1
I
10
II
(3)
III
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
where:
I
II
III
=
=
resistive voltage drop
self and mutual induced BEMF
=
motional BEMF
M*
=
mutual inductance
L*
=
coil self inductance
Substituting the coil self inductance and mutual inductance with an
effective inductance, L, and applying a star connection where
i1 + i2 + i3 = 0:
v1 = i1 R + Li1 + e1 :
v2 = i2 R + Li2 + e2
{ {
v3 = i3 R + Li3 + e{3
I
II
(4)
III
where:
I
II
=
=
resistive voltage drop
self and mutual induced BEMF
III
=
motional BEMF
L
=
coil self inductance
This equation forms the heart of the algorithm. Expression III can
be rewritten as:
e=w
dΨm( Θeff )
dΘ
(5)
where:
e
=
normalized BEMF
ϖ
=
phase inductance
Θ
=
rotor angle
Θeff
=
effective rotor angle for a phase
From (1) and (4) we can express the flux linkage as:
Ψp = Li p + Ψmp
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
(6)
11
SPRA323
where:
Ψp
=
total phase flux linkage
L
=
phase inductance
ip
=
phase current
Ψmp
=
phase flux linkage due to magnet
From (1) we can also derive:
t
Ψ(t ) = ∫ [v (t ) − Ri (t )].dt
(7)
It can be shown that if changes of current and angle are small:
∆Ψ p =
∂Ψp
∂i
∆i p +
∂Ψp
∂Θ p
∆Θ
(8)
This equation describes how total flux linkage changes with
respect to small changes of phase current and angle. We can set
one variable to zero to calculate the other by applying
superposition theory as follows:
∂Ψ p
∂i
= L, and
∂Ψ p
∂Θ
= ep
this gives us:
∆Θ p = −
L
∆i p
ep
∆Ψp = e p ∆Θ
(9)
(10)
As well as electrical equations (9) and (10), the algorithm also
uses a second order mechanical equation of motion. Assuming
that the load of the motor can be expressed as a system
containing torque losses due to friction, torque due to a linear
load, and moments of inertia, we have:
d 2θ e
dθ
I
Te = [ J
+ B e + T1 ]
2
P
dt
dt
12
(11)
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
where:
J
B
=
moment of inertia
=
viscous friction constant
Te
=
resulting torque
T1
=
constant torque
Θ
p
=
rotor angle
=
no. of pairs of poles
The above formulae are used in the algorithm (see Figure 2).
Figure 2. Algorithm Flowchart
Start
Flux linkage
estimation
Rotor Positon
estimation
Current error
calculation
Rotor postiton
correction
Flux linkage
correction
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
13
SPRA323
To summarize, the algorithm consists of the following steps:
1) Estimate the total phase flux linkage using equation (7).
2) Estimate rotor position using the mechanical equation of
motion (10).
3) Calculate current error by comparing measured current with
that calculated using equation (6).
4) Correct position estimate using equation (9).
5) Correct flux linkage estimate using equation (10).
14
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
Simulation Results
As part of a continuing study by Allan into applying integrated
motor design to optimize motor performance, Weber &
Munzenmay developed an implementation of the algorithm written
in C and run on a PC.3 4 Motor voltage and current data was
collected from PSpice models and an experimental trapezoidal
motor. This data was then used offline to determine rotor position
as the algorithm could not run sufficiently fast to operate online
and in real time.
As the offline simulation proved successful, a test setup using a
fast DSP has now been designed to fully prove the algorithm
running in real-time. In addition, the motor commutation is to be
controlled in a closed loop. This study also differs from Ertugrul
and Acamley’s work in that a trapezoidal motor is to be used. The
nature of the trapezoidal motor currents makes this a more difficult
task for the algorithm than for the sinusoidal case.
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
15
SPRA323
Experimental Setup
The experimental setup includes a TI TMS320C25 16-bit fixed
point DSP coprocessor card with the following components:
q
q
q
q
q
q
q
q
q
8K words RAM
12-bit 100 kHz analogue-to-digital converter (ADC)
4 TTL digital inputs
4 TTL digital outputs
4 anti-aliased analogue inputs
Programmable gain amplifier
Dual 12-bit digital-to-analog (DAC) converters
3 16-bit timer/counter channels
16-bit 512 word FIFO interface to the host PC
Other equipment includes:
q
q
q
q
120 MHz Pentium host PC
3 current interfaces
External multiplexer/attenuator
Optical shaft encoder motor drive circuitry
Figure 3 shows the experimental setup. The multiplexer presents
the motor phase currents and voltages to the coprocessor’s ADC.
The inverter is driven by the coprocessor’s digital outputs.
16
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
Figure 3. Experimental Setup
Coprocessor
ADC
input
Host PC
Inverter
control
intputs
BLDCM
Multiplexer
Inverter
Motor v,1
Motor
Power
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
17
SPRA323
Algorithm Implementation
The algorithm was initially coded using the TI C cross compiler,
but this proved to be too slow for motor operation above a few
rpm. This was primarily due to the use of floating point variables.
The algorithm was then coded using the TI cross assembler,
which allowed operation up to around 3000 rpm.
The DSP internal timer was set to provide interrupts every 167 µs.
This rate was chosen because it gives 2° accuracy at 2000 rpm.
Conversion of the seven ADC channels (3 phase currents, 3
phase voltages, and the starpoint voltage) takes 70 µs and the
calculations takes approximately 90 µs.
On receiving a timer interrupt, the DSP proceeds to initiate seven
ADC conversions using polling to eliminate any interrupt timing
overheads. The various calculations are then performed.
The flux estimation based on equation (7) is carried out using the
simple integration algorithm:
Ψp ( k ) = T (v p ( k ) = Ri p ( k ) ) + Ψp ( k −1)
(12)
where:
Ψp ( k )
=
flux at time k
T
=
sampling interval
v p( k )
=
phase voltage at time k
R
=
phase resistance
i p( k )
=
phase current at time k
Ψp ( k −1)
=
flux at time k-1
To calculate the rotor position estimate directly from the
mechanical equation of motion (11) requires considerable
processing time. Because this is a second order function it can be
approximated by a second order polynomial:
θ = at 2 + bt + c
By fitting this polynomial to position data at k − 2 , k − 1 and k , the
position at time k + 1 can be predicted. Let t = 0 at time k − 2 :
18
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
θ k −2 = c
θ k −1 = aT 2 + bT + c
θ k = 4aT 2 + 2bT + c
θ k +1 = 9aT 2 + 3bT + c
Solving:
θ k +1 = 30 k − 30 k −1 + θ k −2
(13)
To obtain the current error, equation (6) is rearranged and
subtracted from the measured current to give:
∆i p = i p −
Ψmp − Ψp
L
(14)
The permanent magnetic flux linkage, Ψmp is obtained from a look
up table.
The rotor position and flux linkage corrections are calculated from
equations (9) and (10). The BEMF value, e p , is obtained from a
look up table.
The above calculations are carried out three times, once per
phase, resulting in three position values. These values are
averaged to provide the final position estimate.
This final position estimate is used to determine when to switch
the motor phase voltages. This is done by writing the appropriate
bit pattern to the digital outputs. These outputs drive power
transistors in the inverter module.
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
19
SPRA323
Experimental Results
As implemented, the algorithm requires the motor BEMF to be
available from a look up table. The BEMF is normalized against
speed to keep the table within reasonable proportions. This
approach is only valid if the BEMF magnitude is linearly
proportional to motor speed. This was tested by driving the test
motor mechanically from another motor and measuring the BEMF
at the motor phase winding terminals. Figure 4 shows the results
and confirm that the BEMF is approximately linear over the
proposed speed range from 1000 rpm to 3000 rpm.
Figure 4. Motor BEMF Magnitude
2.5
♦
♦ ♦
2
♦
BEMF (V)
1.5
♦
1
♦
♦
0.5
0
0
1000
2000
3000
4000
Speed (rpm)
By dividing the BEMF at any one speed by the speed, the
normalized BEMF can be obtained. This was done and the
normalized BEMF was stored in a table at one value per degree of
rotation (see Figure 5).
20
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
Figure 5. Normalized BEMF
1.00E-02
BEMF
5.00E-03
0.00E-00
0
90
180
270
-5.00E-03
-1.00E-02
Rotor position (degrees)
The permanent magnet flux linkage required by the current error
formula is also obtained from a look up table. Integrating the
normalized BEMF data, as shown in Figure 6, produced the look
up table.
Figure 6. Permanent Magnet Flux Linkage
-1.00E-02
Flux (Webers)
-1.00E-02
-1.00E-02
0
90
180
270
-1.00E-02
-1.00E-02
Rotor position (degrees)
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
21
SPRA323
With the motor-specific data in place, the rotor position algorithm
could be tested. This was done by running the motor up to a
constant speed of 2000 rpm using traditional hall effect sensors to
provide commutation information. Once the motor reached a
stable speed, the commutation function was taken over by the
DSP. The motor continued to operate normally. The calculated
rotor position and position error are shown in Figure 7.
Figure 7. Calculated Position and Error
_______Position . . . . . . . . . .Error
400
4
2
300
250
0
200
-2
150
100
Error (degrees)
Calculated position
(degrees)
350
-4
50
-6
0
Time (7.5ms/div)
Although the algorithm is designed to operate over a wide speed
range, including start up, this could not be shown to operate
correctly. The failure to operate at start up appears to be caused
by instability in the current error calculation section, brought on by
spikes in the measured phase currents.
22
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
SPRA323
Summary
Although the algorithm functions correctly under certain stable
operating conditions, problems remain. Most importantly, while it
is possible to start the motor open loop, it is desirable that the
algorithm operate at start up. Additionally, the algorithm has an
inherent limitation in that the initial position of the rotor must be
known before start up. This is necessary to provide the correct
initial values to the position prediction polynomial and ensure that
the correct initial values are retrieved from the BEMF and flux
tables.
Future Work
The possibility of filtering any sudden changes in the current error
needs to be examined. Additionally, the stability of the algorithm
under step changes in load and speed must be fully explored.
Finally, there would be advantages in coding the algorithm for a
floating point DSP, such as the TMS320C30. A significant
increase in processing speed would allow for faster sampling
rates. Coding of experimental algorithm changes would be made
much simpler as variable scaling (due to the fixed point nature of
the TM5320C25) would not be required.
References
1
N. Ertugrul; Position Estimation and Performance Prediction for
Permanent Magnet Motor Drives; PhD Thesis, University
of Newcastle upon Tyne, 1993
2
P.P. Acamley, N. Ertugrul; "Rotor Position Estimation In PM Motors";
ICEM 1992
3
M. Allan, I, Kemp, T. Westwood, B. Pitches; "Application Flexible
Design Strategies For Power Electronic Drives"; PCIM
’90, Munich, West Germany, June 25-29, 1990
4
M. Weber, M. Munzenmay; A Signal Processing Approach To Position
Measurement In Electrical Machines; Diploma Thesis,
Glasgow Caledonian University, January 1996
Sensorless Control of a Trapezoidal Brushless DC Motor Using the TMS320C25 DSP
23